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Area of Trapezium
The area of a trapezium is defined as the number of unit squares that can fit into it and it is measured in square units (like cm2, m2, in2, etc). A trapezium is a four-sided geometric shape that has one pair of parallel sides (generally referred to as "bases"). The other pair of sides of a trapezium can be non-parallel and are known as "legs". The area of a trapezium is the total space covered by a trapezium in a two-dimensional plane.
For example, if 20 unit squares each of length 1 in can be fit inside a trapezium, then its area is 20 in2. It is not always possible to calculate the area of a trapezium by drawing unit squares. Deriving a standard formula for area of trapezium would help. We will learn about the area of a trapezium and the formula to calculate it on this page.
What is Area of Trapezium?
The area of trapezium is calculated by the formula 1/2 × (sum of its parallel sides) × (its height). This formula can be simplified by calculating the average of lengths of the parallel sides (average length) and then it simplifies to (average length) × (height). Here, "height" refers to the perpendicular distance between two parallel sides of the trapezium.
Thus, the area of a trapezium depends upon the lengths of the parallel sides and the distance between them. Let us see how this formula can be derived in different ways in the sections below.
Area of Trapezium Formula
The area of a trapezium can be calculated using the lengths of two of its parallel sides and the distance (height) between them. The formula to calculate the area (A) of a trapezium using base and height is given as,
A = ½ (a + b) h where,
- a and b = bases of the trapezium (parallel sides), and,
- h = height (the perpendicular distance between a and b)
We can derive the formula to find the area of a trapezium in two ways:
- Using a parallelogram
- Using a triangle
Area of Trapezium Derivation Using a Parallelogram
To derive the formula for the area of a trapezium using parallelogram, we will consider two identical trapeziums, each with bases a and b and height h. Let A be the area of each trapezium. Assume that the second trapezium is turned upside down as shown in the figure below.
Now, attach the above two trapeziums.
We can see that the new figure obtained by joining the two trapeziums is a parallelogram whose base is a + b and whose height is h. We know that
- the area of a parallelogram is base × height = (a + b) h.
- the area of the above parallelogram in terms of 'A' is, A + A = 2A.
Thus, 2A = (a + b) h
⇒ A = (a+b)h/2
Thus, the formula for the area of a trapezium is derived.
Area of Trapezium Derivation Using a Triangle
We will derive the area of a trapezium formula by using a triangle here. Consider the above trapezium of bases a and b and height h. In order to derive the formula,
- Step 1: Split one of the legs of the trapezium into two equal parts.
- Step 2: Cut a triangular portion from the trapezium (as shown in the top figure of the below diagram).
- Step 3: Attach it at the bottom (as shown in the bottom figure in the diagram below).
Now, the trapezium is rearranged as a triangle. It can be concluded from the above diagram that the areas of both the trapezium and the triangle are equal. Also, it can be observed that the base of the triangle is equal to (a + b) and the height of the triangle still is h.
The area of the trapezium = The area of the triangle = ½ × base × height = ½ (a + b) h
How To Find Area of Trapezium?
We use the area of a trapezium formula mentioned above to calculate the area. Here is the step-by-step explanation to find the area of a trapezium:
- Step 1: Note the dimensions of the lengths of the parallel sides (bases) of the trapezium from the given data. Make sure that all dimensions are in the same unit.
- Step 2: Calculate the sum of the bases.
- Step 3: Multiply the value of the sum of bases by the height or altitude of the trapezium and by 1/2.
- Step 4: Write the answer in square units.
Area of Trapezium by Heron's Formula
Area of trapezium can be calculated when the coordinates of its vertices are given though its height is NOT given. Assume that a trapezium ABCD is given along with the coordinates of A, B, C, and D. Then draw one of the diagonals (say AC) to divide the trapezium into two triangles and then compute the area of the trapezium as follows:
- Find the lengths of all sides AB, BC, CD, and DA, and also find the length of AC using the distance formula.
- Now, use Heron's formula to compute the areas of triangles ABC and ADC.
- Then add these two areas which will give the area of the total trapezium ABCD.
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Area of Trapezium Examples
Example 1: Find the height of the trapezium if its area is 128 sq in and the lengths of the bases are 18 in and 14 in.
Lengths of bases of trapezium = 18 in and 14 in
Area = 128 sq in
Let the height of the trapezium be 'h' units.
Using the trapezium formula, we know, Area = ½ (Sum of bases) height
Substitute all these values in the area of the trapezium formula,
128 = ½ (18 + 14) h
⇒ h = 128 × 2/32
⇒ h = 8 in
Answer: The height of the given trapezium = 8 in.
Example 2: Find the area of an isosceles trapezium with the length of each leg to be 13 units and the bases of lengths 22 units and 12 units.
Bases, b = 22 units; a = 12 units.
Let us assume that its height is h. We can divide the given trapezium into two congruent right triangles and a rectangle as follows:
From the above figure,
x + x + 12 = 22
⇒ 2x + 12 = 22
⇒ 2x = 10
⇒ x = 5
Using Pythagoras theorem,
x2 + h2 = 132
⇒ 52 + h2 = 169
⇒ 25 + h2 = 169
⇒ h2 = 144
⇒ h = √144 = 12
The area of the given trapezium is,
A = ½ (a + b) h
⇒ A = ½ (12 + 22) (12) = 204 square units
Answer: The area of the given trapezium = 204 square units.
Example 3: Find the area of a trapezium whose bases are 25 in and 10 in and whose height is 6 in.
The bases are a = 25 in; b = 10 in.
The height is h = 6 in.
The area of the trapezium is, A = ½ (a + b) h
⇒ A = ½ (25 + 10) (6) = ½ (35) (6) = 105 in2.
Answer: Area of the given trapezium is 105 in2.
Practice Questions on Area of Trapezium
FAQs on Area of Trapezium
What is the Definition of Area of a Trapezium?
The area of a trapezium is the total space covered by the shape in the two-dimensional plane. To calculate the area of a trapezium, we simply multiply the sum of bases by the height and divide the obtained product by 2.
What is the Area of Trapezium Formula?
The area of a trapezium is calculated using the formula 1/2 · (a + b) · h, where
- 'a' and 'b' are the lengths of the parallel sides and
- 'h' is the height of the trapezium
How to Calculate the Area of Trapezium Without Height?
If the height of the trapezium is not given and all its sides are given instead, then we will divide it into two right triangles and a rectangle. The areas of each of these shapes can be calculated and added to find the area of the given trapezium.
How to Calculate the Area of Trapezium Using Bases and Height?
The area of a trapezium, given the height and bases, can be calculated by multiplying the average of the base lengths by the height of the trapezium. The height of a trapezium is simply the distance between the parallel sides.
What are the Formulas for Perimeter and Area of Trapezium?
The area of a trapezium can be calculated using the formula: A = ½ × (a + b) × h. While the formula to calculate the perimeter of the trapezium is given as: P = a + b + c + d.
- a and b are the parallel sides
- c and d are the non-parallel sides
- h = Distance between the two parallel sides i.e., a and b.
What is the Difference Between Trapezium and Trapezoid?
The words trapezium and trapezoid are often used interchangeably. But these two terms mean different depending on the geography. In the United States and Canada, a trapezoid is a quadrilateral with at least one pair of parallel sides is trapezoid whereas a quadrilateral with no parallel sides (basically an irregular quadrilateral) if a trapezium. Outside the US and Canada, these two definitions are interchanged.
How do you Prove the Area of a Trapezium Formula?
The area of a trapezium formula can be proved by rearranging the trapezium in form of a parallelogram or a triangle. The area of the shapes thus obtained can be related to a trapezium for proving the area of a trapezium formula.
What is the Area of the Trapezium Formula Using the Median?
The "median" of a trapezium refers to the line segment that joins the midpoints of the non-parallel sides of the trapezium. The "height" of the trapezium is the perpendicular distance between the two parallel sides. Then the area of trapezium = median × height. Note that this formula doesn't need the lengths of the parallel sides (bases).
What is the Use of the Trapezium Area Calculator?
Area of trapezium calculator is used to calculate the area of the trapezium by entering the given specific parameters. Such as height and the values of the bases. It gives the measurement easily and quickly. Try area of trapezium calculator and get your answers with just a click.
How do you Find the Area of an Isosceles Trapezium Using Sides?
The area of an isosceles trapezium can be calculated if the measurements of its sides are known. For this, we can divide the shape in the form of a rectangle and 2 right-angled triangles. The height can be calculated by applying Pythagoras theorem in one of the right-angled triangles and finally, we can apply the area of the trapezium formula to calculate the required area.